Probability Review Answers (a) (0.45, 45 %) (A1)(A1) (C2) 200 Note: Award (A1) for numerator, (A1) for denominator.

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1 Probability Review Answers 90. (a) (0.45, 45 %) (A)(A) (C) 00 Note: Award (A) for numerator, (A) for denominator. 60 (b) ( 0.6, 0.667, 66.6%, %, 66.7 %) (A)(A)(ft) (C) 90 Notes: Award (A) for numerator, (A)(ft) for denominator, follow through from their numerator in part (a). Last mark is lost if answer is not a probability. (c) Note: Award (M) for correct substitution in the combined events formula. Follow through from their answer to part (a). (M) 0 = (0.65, 65 %) (A)(ft) 00 OR Note: Award (M) for adding the correct fractions. (M) 0 = (0.65, 65 %) (A) 00 OR = (M) 00 Note: Award (M) for subtraction of correct fraction from. 0 = (0.65, 65%) (A) (C) 00 [6] IB Questionbank Mathematical Studies rd edition

2 . (a) (A) for rectangle and three intersecting circles (A) for 0, (A) for 8, and 9, (A) for, 5 and 7 (A4) (b) 00 ( ) = 6 (M)(A)(ft)(G) Note: Follow through from their diagram. 5 (c) (0.5) (A)(ft) 00 = 5% (A)(ft)(G) Note: Follow through from their diagram. (d) Note: The following statements are correct. Please note that the connectives are important. It is not the same (had cereal) and (not bread) and (had cereal) or (not bread). The parentheses are not needed but are there to facilitate the understanding of the propositions. (had cereal) and (did not have bread) (had cereal only) or (had cereal and fruit only) (had either cereal or (fruit and cereal)) and (did not have bread) (A)(A) Notes: If the statements are correct but the connectives are wrong then award at most (A)(A0). For the statement (had only cereal) and (cereal and fruit) award (A)(A0). For the statement had cereal and fruit award (A0)(A0). IB Questionbank Mathematical Studies rd edition

3 54 (e) (0.54, 54 %) (A)(ft)(A)(ft)(G) 00 Note: Award (A)(ft) for numerator, follow through from their diagram, (A)(ft) for denominator. Follow through from total or denominator used in part (c). 0 9 (f) ( , 0.909%) (A)(ft)(M)(A)(ft)(G) Notes: Award (A)(ft) for their correct fractions, (M) for multiplying two fractions, (A)(ft) for their correct answer. Answer with no working receives no marks. Follow through from denominator in parts (c) and (e) and from their diagram. [5]. (a) Note: Award (A) for each correct pair. (A)(A)(A) (C) (b) = 0.07 (, 7%) (A)(ft) (C) 00 (c) (M) = 0. (, %) 00 (A)(ft) (C) Note: In (b) and (c) follow through from sensible answers only i.e. not a probability greater than one. [6] IB Questionbank Mathematical Studies rd edition

4 4. (a) (A)(A)(A) (C) Note: Award (A) for a labeled Venn diagram with appropriate sets. (A) for 7, (A) for 8 and (b) P (Spanish / one language only) = (M)(A)(ft) Note: Award (M) for substituted conditional probability formula, (A) for correct substitution. Follow through from their Venn diagram. 8 = (0.65, 6.5%) (A)(ft) OR P (Spanish / one language only) = (A)(ft)(M) Note: Award (A) for their correct numerator, (M) for correct recognition of regions. Follow through from their Venn diagram. 8 = (0.65, 6.5%) (A)(ft) (C) [6] IB Questionbank Mathematical Studies rd edition 4

5 5. (a) (A5) 5 Note: Award (A) for 0.5 at B, (A) for 0. at T, then (A) for each correct pair. Accept fractions or percentages. (b) 0.06 (A)(ft) Note: Accept or 6% (c) for a relevant two-factor product, either C L or T L for summing three two-factor products ( ) (M)(M) (A)(ft)(G) (d) (M)(A)(ft) Note: Award (M) for substituted conditional probability formula seen, (A)(ft) for correct substitution = 0.55 (A)(ft)(G) [] IB Questionbank Mathematical Studies rd edition 5

6 6. (a) (A) (A) (A) (A) Note: Award (A) for correct tree structure, (A) for each complementary pair. IB Questionbank Mathematical Studies rd edition 6

7 (b) (i) (M) 0 (c) (ii) 5 0 (0.05, 5%) 0 Note: Award (M) for correct product ,(0.667) Note: Award (M) for finding two products, (M) for adding two products. (0.5,5%) Note: Award (M) for using the conditional probability formula. Note: Award (A) for correct numerator, (A) for correct denominator. (A)(ft)(G) (M)(M) (A)(ft)(G) (M) (A)(A)(ft) or (G) [] 7. (a) (A4) (C4) Note: Award (A4) for all 5 correct, (A) for 4 correct, (A) for correct, (A) for correct. IB Questionbank Mathematical Studies rd edition 7

8 (b) (M)(M) Note: Award (M) for (M) for , (M) for adding (M) for answer. = (A) = 0.55 (A) (C4) [8] 8. (a) For solving for P(A B) from the formula in their tables (M) P(A B) = 0. (A) (C) (b) Because need to see the numbers, not just a statement (R) Therefore no, not independent (A) (C) Note: Cannot award (A) if (R) not awarded. (c) Because P(A B) 0 (R) Not mutually exclusive (A) (C) Note: Cannot award (A) if (R) not awarded. [6] 9. (a) a = 4, b = (A)(A) (b) 0 ( ) = (M)(A) (or (A)) (c) (A)(A) Note: Award (A) for numerator, (A) for denominator. (d) 6 9 (A)(A) Note: Award (A) for numerator, (A) for denominator. [8] IB Questionbank Mathematical Studies rd edition 8

9 0. U S C D 5 (a) intersecting circles and rectangle. (A) correct numbers (A4) 5 Note: Award (A4) for all 7 numbers correct, (A) for 6 correct, (A) for 5 correct, (A) for 4 correct. (Do not count the number in D only on the Venn Diagram at this stage.) (b) x = 40 (M) 5 + x = 40 x = 5 Therefore, five play drums only. (A) (AG) (c) 4,0%, (A) Note: Award (A) for 4, (A) for 40. (d) (5.5%, 0.55) 40 (A) Note: Award (A) for, (A) for 40. (e) 8,50%, (A) Note: Award (A) for 8, (A) for 6. Do not separate (A). [4] IB Questionbank Mathematical Studies rd edition 9

10 . (a) 0.98 t a 0.95 p not p b 0.55 not t t a = 0.05 b = 0.0 c = 0.45 c not t (A) (A) (A) (C) (b) = 0.9 (M)(A) (C) (c) (M)(M) = (A) (C) [8]. (a) U W C 6 8 F (A4) 4 Note: Award (A) for the box and circles, (A) for 7 correct entries, (A) for 5 or 6 correct, (A) for or 4 correct. (b) 00 ( ) = (M)(A) IB Questionbank Mathematical Studies rd edition 0

11 7 (c) (i) P(coffee) = (= 0.7) (A) 00 7 (ii) P(W F C) = (= 0.7) (A) 00 8 (iii) P(FW) = (= 0.58) (A)(A) 4 Note: Award (A) for 8, (A) for. 5 4 (d) P(both drank all ) = (A)(A) = (= 0.000) (A) 9900 []. (a) (A) (C) (b) 4 or 5% (A) (C) (c) 6 or 6.7% ( s.f.) (A) (C) (d) 4 or.% ( s.f.) (A) (C) [8] 4. (a) Boy Boy Girl Girl Boy Girl (A) Note: Award (A) for 5 or 4 correct probabilities, (A) for or, (A0) for or 0. (b) (i) P( boys) = (A) 4 IB Questionbank Mathematical Studies rd edition

12 (ii) P( boys first child is boy) = 4 (A) (iii) P( boys boy in family) = 4 (A) 4 Note: Answers can be obtained logically without using conditional probability. Award full marks for correct answers. [5] 5. (a) 0. F A B NF F 0.88 NF Note: Award (A) for two or three bold entries correct. (A) (b) (i) = 0.06 (A) (ii) = 0.89 (M)(A) [4] IB Questionbank Mathematical Studies rd edition

13 6. (a) (i) D S U x 48 x 44 x (A) Note: Award (A) for a correct diagram (labelled), (A) for x in the correct position, (A) for either (48 x) or (44 x) correctly positioned. (ii) 48 x + x + 44 x = 60 (or equivalent), allow ft from (i) (M) x = (A) (iii) The set of members who did not attend for both (A) Drama and Sports (or equivalent) (iv) P(D or S) = (M)(M) Note: Award (M) for either or, (M) for adding. 8 7 = or or ( s.f.) or 46.7% (s.f.) (A) (b) (i) P(Female and (S or D)) = (M) 60 = or 0. ( s.f.) or.% (s.f.) (A) 8 (ii) P(Male and both D and S) = (M) 60 = or 0.4 or 40% (A) 5 [4] 7. (a) IB Questionbank Mathematical Studies rd edition

14 5 B (A) J 5 W J B (A) W (b) P(J W) =, P(J W) = (M) 5 Note: Award (M) for either correct. P(W) = (M) = or ( s.f.) or 46.7% ( s.f.) (A) 5 (c) P(J W W) =, P(J W W) = 0 (M) 5 4 P(W W) = 0 0 = or 0.5 or 5% (A) 0 [7] 8. (a) (M) = 0.65 accept 6.5% or 00 (A) IB Questionbank Mathematical Studies rd edition 4

15 (b) (M) 9 accept 4.5% or = (A) [4] 9. (a) Study Spanish Do not study Spanish Total Have computers Do not have computers 4 7 Total 0 5 (A) Notes: Award ½ mark for each correct bold entry. Round up to a maximum marks. (b) S 7 8 C U 4 (A) Notes: Award mark for correctly labelled diagram indicating all of S, C and U. Award ½ mark for any two correct entries from, 7, 8 and 4, up to a maximum of mark. (c) Students who have computers or do not study Spanish (or equivalent statements) (C) Notes: Award (R0) for Students who have computers but do not study Spanish. It is incorrect. (d) n(c S) = 9 (A) IB Questionbank Mathematical Studies rd edition 5

16 7 (e) (i) (M)(A) 5 8 (ii) (M)(A) 5 7 (iii) (M)(A) 0 Notes: If no method is shown, award (M)(A) if and only if answer is correct, otherwise award zero marks. However, award (M) if correct method is shown; even if final answer is wrong. Accept probabilities expressed as decimals or percentages. [4] 0. 4 Red 5 Red 4 Green 5 Green 4 Red 4 Green Note: Award (A) for each correct bold entry (A4) (C4) [4] IB Questionbank Mathematical Studies rd edition 6